Reports de recerca
http://hdl.handle.net/2117/3919
Thu, 08 Dec 2016 06:08:03 GMT2016-12-08T06:08:03ZConnected and internal graph searching
http://hdl.handle.net/2117/97422
Connected and internal graph searching
Barrière Figueroa, Eulalia; Fraigniaud, Pierre; Santoro, Nicola; Thilikos Touloupas, Dimitrios
This paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.
Tue, 29 Nov 2016 13:30:39 GMThttp://hdl.handle.net/2117/974222016-11-29T13:30:39ZBarrière Figueroa, EulaliaFraigniaud, PierreSantoro, NicolaThilikos Touloupas, DimitriosThis paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.Unranking of combinatorial structures
http://hdl.handle.net/2117/97301
Unranking of combinatorial structures
Molinero Albareda, Xavier
The present work is a study of the deterministic generation of
combinatorial structures. We generate classes of combinatorial
structures formally specifiable by grammars involving unlabelled
sequence, unlabelled set, unlabelled multiset, unlabelled pointing,
unlabelled unpointing, unlabelled substitution, labelled sequence,
labelled set, labelled multiset, labelled cycle, labelled pointing,
labelled unpointing and labelled substitution constructions.
It consists of some basic preliminaries about classes
of combinatorial structures and three chapters. The first
one, Unlabelled Unranking , solves the unranking of unlabelled
structures. The second one, Labelled
Unranking, solves the unranking of labelled
structures. The third one, Product of labellings,
explains the order of the labellings in the structures. Appendix
contains the libraries developed in this work.
Mon, 28 Nov 2016 10:45:06 GMThttp://hdl.handle.net/2117/973012016-11-28T10:45:06ZMolinero Albareda, XavierThe present work is a study of the deterministic generation of
combinatorial structures. We generate classes of combinatorial
structures formally specifiable by grammars involving unlabelled
sequence, unlabelled set, unlabelled multiset, unlabelled pointing,
unlabelled unpointing, unlabelled substitution, labelled sequence,
labelled set, labelled multiset, labelled cycle, labelled pointing,
labelled unpointing and labelled substitution constructions.
It consists of some basic preliminaries about classes
of combinatorial structures and three chapters. The first
one, Unlabelled Unranking , solves the unranking of unlabelled
structures. The second one, Labelled
Unranking, solves the unranking of labelled
structures. The third one, Product of labellings,
explains the order of the labellings in the structures. Appendix
contains the libraries developed in this work.The Plogi and ACi-1 operators on the polynomial time hierarchy
http://hdl.handle.net/2117/97278
The Plogi and ACi-1 operators on the polynomial time hierarchy
Castro Rabal, Jorge; Seara Ojea, Carlos
In a previous paper ([CS-92]) we studied the agreement of operators P_{log^i} and AC^{i-1} acting on NP. In this article we extend this work to other classes of the polynomial time hierarchy. We show that on Sigma_k^p, Pi_k^p, Delta_k^P and Theta_k^P-classes both operators have the same behaviour, but this coincidence does not seem to be true on other classes included in the PH hierarchy: we give a set A such that, relativized to A, P_{log^i}(P_{log^j}(NP)) is different from AC^{i-1}(P_{log^j}(NP)). As a result of these characterizations we show P_{log}(Theta_k^p) = Theta_k^p, an equality that is useful to show lowness properties. In fact, we get easily the Theta-lowness results given by Long and Sheu in their paper [LS-91]. Besides, we clarify the situation of the classes in L_2^{p,Delta} for which their membership to L_2^{p,Theta} was not clear.
Mon, 28 Nov 2016 09:01:43 GMThttp://hdl.handle.net/2117/972782016-11-28T09:01:43ZCastro Rabal, JorgeSeara Ojea, CarlosIn a previous paper ([CS-92]) we studied the agreement of operators P_{log^i} and AC^{i-1} acting on NP. In this article we extend this work to other classes of the polynomial time hierarchy. We show that on Sigma_k^p, Pi_k^p, Delta_k^P and Theta_k^P-classes both operators have the same behaviour, but this coincidence does not seem to be true on other classes included in the PH hierarchy: we give a set A such that, relativized to A, P_{log^i}(P_{log^j}(NP)) is different from AC^{i-1}(P_{log^j}(NP)). As a result of these characterizations we show P_{log}(Theta_k^p) = Theta_k^p, an equality that is useful to show lowness properties. In fact, we get easily the Theta-lowness results given by Long and Sheu in their paper [LS-91]. Besides, we clarify the situation of the classes in L_2^{p,Delta} for which their membership to L_2^{p,Theta} was not clear.Generic algorithms for the generation of combinatorial objects
http://hdl.handle.net/2117/97031
Generic algorithms for the generation of combinatorial objects
Molinero Albareda, Xavier; Martínez Parra, Conrado
This report briefly describes our generic approach to the exhaustive generation of unlabelled and labelled combinatorial classes. Our algorithms receive a size n and a finite description of a combinatorial class A using combinatorial operators such as union, product, set or sequence, in order to list all objects of size n in A. The algorithms work in constant amortized time per generated object and thus they are suitable for rapid prototyping or for inclusion in general libraries.
Tue, 22 Nov 2016 14:37:57 GMThttp://hdl.handle.net/2117/970312016-11-22T14:37:57ZMolinero Albareda, XavierMartínez Parra, ConradoThis report briefly describes our generic approach to the exhaustive generation of unlabelled and labelled combinatorial classes. Our algorithms receive a size n and a finite description of a combinatorial class A using combinatorial operators such as union, product, set or sequence, in order to list all objects of size n in A. The algorithms work in constant amortized time per generated object and thus they are suitable for rapid prototyping or for inclusion in general libraries.An Efficient generic algorithm for the generation of unlabelled cycles
http://hdl.handle.net/2117/97029
An Efficient generic algorithm for the generation of unlabelled cycles
Martínez Parra, Conrado; Molinero Albareda, Xavier
In this report we combine two recent generation algorithms to obtain a
new algorithm for the generation of unlabelled cycles. Sawada's
algorithm lists all k-ary unlabelled cycles with fixed
content, that is, the
number of occurences of each symbol is fixed and given a priori.
The other algorithm, by the authors, generates all
multisets of objects with given total size n from any admissible
unlabelled class A. By admissible
we mean that the class can be specificied using atomic classes,
disjoints unions, products, sequences, (multi)sets, etc.
The resulting algorithm, which is the main contribution of this paper,
generates all cycles of objects with given total size n from any
admissible class A. Given the
generic nature of the algorithm, it is suitable for inclusion in
combinatorial libraries and for rapid prototyping. The new algorithm
incurs constant amortized time per generated cycle, the constant only
depending in the class A to which the objects in the cycle belong.
Tue, 22 Nov 2016 14:33:29 GMThttp://hdl.handle.net/2117/970292016-11-22T14:33:29ZMartínez Parra, ConradoMolinero Albareda, XavierIn this report we combine two recent generation algorithms to obtain a
new algorithm for the generation of unlabelled cycles. Sawada's
algorithm lists all k-ary unlabelled cycles with fixed
content, that is, the
number of occurences of each symbol is fixed and given a priori.
The other algorithm, by the authors, generates all
multisets of objects with given total size n from any admissible
unlabelled class A. By admissible
we mean that the class can be specificied using atomic classes,
disjoints unions, products, sequences, (multi)sets, etc.
The resulting algorithm, which is the main contribution of this paper,
generates all cycles of objects with given total size n from any
admissible class A. Given the
generic nature of the algorithm, it is suitable for inclusion in
combinatorial libraries and for rapid prototyping. The new algorithm
incurs constant amortized time per generated cycle, the constant only
depending in the class A to which the objects in the cycle belong.General bounds on limited broadcast domination
http://hdl.handle.net/2117/96711
General bounds on limited broadcast domination
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz; Cáceres, José
Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded by a constant k . The minimum cost of such a dominating broadcast is the k -broadcast dominating number. We present a uni ed upper bound on this parameter for any value of k in terms of both k and the order of the graph. For the speci c case of the 2-broadcast dominating number, we show that this bound is tight for graphs as large as desired. We also study the family of caterpillars, providing a smaller upper bound, which is attained by a set of such graphs with unbounded order.
Wed, 16 Nov 2016 10:37:47 GMThttp://hdl.handle.net/2117/967112016-11-16T10:37:47ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelPuertas, Maria LuzCáceres, JoséLimited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded by a constant k . The minimum cost of such a dominating broadcast is the k -broadcast dominating number. We present a uni ed upper bound on this parameter for any value of k in terms of both k and the order of the graph. For the speci c case of the 2-broadcast dominating number, we show that this bound is tight for graphs as large as desired. We also study the family of caterpillars, providing a smaller upper bound, which is attained by a set of such graphs with unbounded order.New results on metric-locating-dominating sets of graphs
http://hdl.handle.net/2117/96709
New results on metric-locating-dominating sets of graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; González, Antonio
A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so far
Wed, 16 Nov 2016 10:25:27 GMThttp://hdl.handle.net/2117/967092016-11-16T10:25:27ZHernando Martín, María del CarmenMora Giné, MercèGonzález, AntonioA dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so farNon-commutative integrable systems on b-symplectic manifolds
http://hdl.handle.net/2117/88884
Non-commutative integrable systems on b-symplectic manifolds
Miranda Galcerán, Eva; Kiesenhofer, Anna
In this paper we study non-commutative integrable systems on
b-Poisson manifolds. One important source of examples (and motiva-
tion) of such systems comes from considering non-commutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and we prove an action-angle theorem for non-commutative integrable systems on a b-symplectic manifold in a neighbourhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure
Tue, 19 Jul 2016 09:30:44 GMThttp://hdl.handle.net/2117/888842016-07-19T09:30:44ZMiranda Galcerán, EvaKiesenhofer, AnnaIn this paper we study non-commutative integrable systems on
b-Poisson manifolds. One important source of examples (and motiva-
tion) of such systems comes from considering non-commutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and we prove an action-angle theorem for non-commutative integrable systems on a b-symplectic manifold in a neighbourhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structureThe HOM problem is decidable
http://hdl.handle.net/2117/87972
The HOM problem is decidable
Godoy, Guillem; Giménez, Omer; Ramos Garrido, Lander; Álvarez Faura, M. del Carme
We close affirmatively a question which has been open for 35 years: decidability of the HOM problem. The HOM problem consists in deciding, given a tree homomorphism $H$ and a regular tree languagle $L$ represented by a tree automaton, whether $H(L)$ is regular. For deciding the HOM problem, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular languages through tree homomorphisms. Our contributions are based on the following new results. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with disequality constraints recognizing the complementary language. We also define a new class of automaton with arbitrary disequality constraints and a particular kind of equality constraints. This new class essentially recognizes the intersection of a tree automaton with disequality constraints and the image of a regular language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism. The above constructions are combined adequately to provide an algorithm deciding the HOM problem.
Tue, 14 Jun 2016 11:51:17 GMThttp://hdl.handle.net/2117/879722016-06-14T11:51:17ZGodoy, GuillemGiménez, OmerRamos Garrido, LanderÁlvarez Faura, M. del CarmeWe close affirmatively a question which has been open for 35 years: decidability of the HOM problem. The HOM problem consists in deciding, given a tree homomorphism $H$ and a regular tree languagle $L$ represented by a tree automaton, whether $H(L)$ is regular. For deciding the HOM problem, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular languages through tree homomorphisms. Our contributions are based on the following new results. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with disequality constraints recognizing the complementary language. We also define a new class of automaton with arbitrary disequality constraints and a particular kind of equality constraints. This new class essentially recognizes the intersection of a tree automaton with disequality constraints and the image of a regular language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism. The above constructions are combined adequately to provide an algorithm deciding the HOM problem.Some advances in the theory of voting systems based on experimental algorithms
http://hdl.handle.net/2117/87345
Some advances in the theory of voting systems based on experimental algorithms
Freixas Bosch, Josep; Molinero Albareda, Xavier
In voting systems, game theory, switching functions, threshold logic, hypergraphs or coherent structures there is an important problem that consists in determining the weightedness of a voting system by means of trades among voters in sets of coalitions. The fundamental theorem by Taylor and Zwicker cite{TaZw92} establishes the equivalence between weighted voting games and $k$-trade robust games for each positive integer $k$. Moreover, they also construct, in cite{TaZw95}, a succession of games $G_k$ based on magic squares which are $(k-1)$-trade robust but not $k$-trade robust, each one of these games $G_k$ has $k^2$ players. The goal of this paper is to provide improvements by means of different experiments to the problem described above. In particular, we will classify all complete games (a basic class of games) of less than eight players according to they are: a weighted voting game or a game which is $(k-1)$-trade robust but not $k$-trade robust for all values of $k$. As a consequence it will we showed the existence of games with less than $k^2$ players which are $(k-1)$-trade robust but not $k$-trade robust. We want to point out that the classifications obtained in this paper by means of experiments are new in the mentioned fields.
Thu, 26 May 2016 08:32:43 GMThttp://hdl.handle.net/2117/873452016-05-26T08:32:43ZFreixas Bosch, JosepMolinero Albareda, XavierIn voting systems, game theory, switching functions, threshold logic, hypergraphs or coherent structures there is an important problem that consists in determining the weightedness of a voting system by means of trades among voters in sets of coalitions. The fundamental theorem by Taylor and Zwicker cite{TaZw92} establishes the equivalence between weighted voting games and $k$-trade robust games for each positive integer $k$. Moreover, they also construct, in cite{TaZw95}, a succession of games $G_k$ based on magic squares which are $(k-1)$-trade robust but not $k$-trade robust, each one of these games $G_k$ has $k^2$ players. The goal of this paper is to provide improvements by means of different experiments to the problem described above. In particular, we will classify all complete games (a basic class of games) of less than eight players according to they are: a weighted voting game or a game which is $(k-1)$-trade robust but not $k$-trade robust for all values of $k$. As a consequence it will we showed the existence of games with less than $k^2$ players which are $(k-1)$-trade robust but not $k$-trade robust. We want to point out that the classifications obtained in this paper by means of experiments are new in the mentioned fields.